This paper studies the generalized Riemann problems for the pressureless flow, where the initial conditions are characterized by piecewise constant densities and piecewise linear velocities. Utilizing characteristic analysis, this research constructs global solutions that include non-constant states, vacuum states, contact discontinuities, and delta-shock waves. The distinctive feature of these generalized Riemann problems are that they permit global solutions which can be explicitly determined.

The object of the present paper is to study paraK\"{a}hler space-time in a perfect fluid. Also, we have studied weakly symmetric and weakly Ricci symmetric perfect fluids in paraK\"{a}hler space-time. Next, we have investigated some curvature identities in paraK\"{a}hler space-time which are pseudo-quasi-conformally flat, pseudo-projectively flat, generalised $W_{2}$-flat, and Bochner flat. Moreover, we have studied some certain curvature properties of paraK\"{a}hler manifold admitting space-time. Furthermore, we have enriched the important properties related to sectional curvature in paraK\"{a}hler space-time.

In this article, we establish some results of homomorphisms and direct products of principal $p$-algebras. We introduce and study the notions of homomorphisms of principal $p$-algebras and triple homomorphisms of principal triples. A relationship between homomorphisms of principal $p$-algebras and triple homomorphisms of principal triples is obtained. We solve some "fill-in" problems concerning principal triples. Finally, we discuss some special filters (ideals) of such algebras with respect to homomorphisms and direct products.

In this article we introduce the notion of deferred sequences spaces defined by a modulus. Using this concept we present a series of inclusion theorems that characterize the relationship between a modulus function and regular summability methods and present theorems as follows: Let $A=[a_{n,k}]$ be a non-negative matrix such that $\sup_{n}\sum_{k=1}^{\infty}a_{n,k}<\infty$ and $f$ is a Modulus function. Then $W^{D}(A,f)\subset W^{D}_{\infty}(A,f)$ and $W^{D}_{0}(A,f)\subset W^{D}_{\infty}(A,f)$.

In this paper, we define the notions of prime, strongly prime and semiprime pseudo symmetric ideals of a partially ordered ternary semigroup. We studied the interesting properties of these special types of pseudo symmetric ideals and proved that the set of all strongly prime pseudo symmetric ideals is topologized. We also proved that the set of all strongly prime pseudo symmetric ideals is a compact space.

The target of this paper is to discuss a new subclass $TS_{\mu,b}^{\alpha}(\hbar,\sigma,\varsigma)$ of univalent functions with negative coefficients related to Hurwitz Lerch Zeta function in the unit disk $ \mathbb { U }=\left \{{z \in \mathbb{ C }:|z|<1}\right \}.$ We obtain basic properties like coefficient inequality, distortion and covering theorem, radii of starlikeness, convexity and close-to-convexity, extreme points, Hadamard product, and closure theorems for functions belonging to our class.

D. Molodtsov introduced the soft set theory in 1999 as a comprehensive mathematical framework for handling uncertain data. Many academics are now using soft set theory to solve decision-making problems. In graph theory, a directed graph is a graph made up of a set of vertices connected by directed edges, also known as arcs. Directed graphs can be used to analyze and solve electrical circuits, shortest routes, social ties and many other problems. J. Jose, B. George and R.K. Thumbakara introduced soft directed graphs by applying the notion of the soft set to directed graphs. Soft directed graphs give a parameterized point of view for directed graphs. In this paper, we define $sd$-walk, $sd$-trail, $sd$-path, $sd$-cycle, directed distance etc. in soft directed graphs. We also introduce multiple types of connectedness, $sd$-radius, $sd$-diameter, and $sd$-center and study some of their features. We prove that a directed graph is a soft directed graph of itself. 

In this article, the concept of semi-symmetry for Lorentz-Sasakian space forms on some $W-$curvature tensors is investigated. Firstly, flatness and $\xi -$flatness states of some $W-$curvature tensors are investigated. Then, characterization of Lorentz-Sasakian space forms in case of semi-symmetry is obtained for the $W-$curvature tensors considered.

Linear recurring sequences over semirings have been introduced recently. This paper is an extension of previous works done in [12]. Certains rectifications and generalizations have been made. In this paper, we study the notion of linear and multilinear recurring sequences over zerosum semirings. After investigating some basics properties, we provide a new characterization of linear and multilinear recurring sequences over zerosum semirings.

For a monoid $S$, the set $S\times S$ equipped with the componentwise right $S$-action is referred to as the diagonal act of $S$ and is denoted by $D(S)$. We explore the preservation of the flatness properties under the diagonal acts. Let $\mathcal{P}$ be some flatness property, $S$ and $T$ be two monoids. It has been shown that both the diagonal $S$-act $D(S)$ and the diagonal $T$-act $D(T)$ preserve property $\mathcal{P}$ if and only if the diagonal $(S\times T)$-act $D(S\times T)$ preserves property $\mathcal{P}$.

Let $R$ be a ring and $\mathscr{C}$ be a class of some finitely presented left $R$-modules. A a left $R$-module $M$ is called \emph{$\mathscr{C}$-injective} if Ext$^1_R(C, M)=0$ for every $C\in \mathscr{C}$; a right $R$-module $M$ is called $\mathscr{C}$-flat, if Tor$^R_1(M, C)=0$ for each $C\in \mathscr{C}$. $R$ is called left $\mathscr{C}$-coherent if every $C\in \mathscr{C}$ is 2-presented; $R$ is called left strongly $\mathscr{C}$-coherent, if whenever $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ is exact, where $C\in \mathscr{C}$ and $P$ is finitely generated projective, then $K$ is $\mathscr{C}$-projective, where a left $R$-module $M$ is called $\mathscr{C}$-projective if ${\rm Ext}^1_R(M, N)=0$ for any $\mathscr{C}$-injective module $N$. In this paper, we give a new characterization of left $\mathscr{C}$-coherent rings. Moreover, when $R$ is a left strongly $\mathscr{C}$-coherent ring, the envelopes and covers by modules of finite $\mathscr{C}$-injective and $\mathscr{C}$-flat dimensions are studied.